Tessellation

A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.

thumb|Potential energy surface for silver depositing on an aluminium–[[palladium–manganese (Al–Pd–Mn) quasicrystal surface. Similar to Fig. 6 in Ref.]]

an infinite array of discrete points in three dimensional space generated by a set of discrete translation operations

thumb|Houndstooth pattern Houndstooth is a pattern of alternating light and dark checks used on fabric. It is also known as hounds tooth check, '''hound's tooth (and similar spellings), dogstooth, dogtooth or dog's tooth'''. The duotone pattern is characterized by a tessellation of light and dark solid checks alternating with light-and-dark diagonally-striped checks—similar in pattern to gingham plaid but with diagonally-striped squares in place of gingham's blended-tone squares. Traditionally, houndstooth uses black and white, although other contrasting colour combinations may be used.

five tiles used in Islamic decorative art

thumb|This beach ball would be a hosohedron with 6 [[spherical lune faces, if the 2 white caps on the ends were removed and the lunes extended to meet at the poles.]]

subdivision of the plane into polygons that are all regular

tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions

a tiling of the plane by pentagons

tessellation of n-dimensional Euclidean space by congruent parallelotopes

Group realized geometrically by reflections across the sides of a triangle

thumb|upright=1.35|This form of the aperiodic tiling|aperiodic [[Penrose tiling has two prototiles, a thick rhombus (shown blue in the figure) and a thin rhombus (green).]] In mathematics, a prototile is one of the shapes of a tile in a tessellation.

thumb|200px|The "sphinx" polyiamond rep-tile. Four copies of the sphinx can be put together as shown to make a larger sphinx. In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of Scientific American. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.

method to quickly identify geometric figures capable of tessellation

on surrounding polygons by layers of copies

Square tiles used in graphic design

type of puzzle

On lattices and sphere packing in Euclidean space

Set of shapes that can be tiled with smaller replicas of the same set

rectangular tilings using various shapes of rectangles

conjecture in geometry about hypercube tiling